What’s Next?
the mathematical legacy of Bill Thurston

Titles and Abstracts

Panel Discussion

Communicating mathematics

Bill Thurston was very interested in how it is that mathematical ideas are communicated from one person to another. The purpose of this panel is to explore this question. The panelists will be Benson Farb, Etienne Ghys, Curt McMullen, Steve Strogatz and Jeff Weeks.

Ian Agol, University of California at Berkeley

The malnormal special quotient theorem

We’ll describe a new proof of Dani Wise’s
Malnormal Special Quotient Theorem, which
states that a special cubulated hyperbolic group may
be filled along malnormal quasiconvex subgroups
to get virtually special quotients. This is a key
technical result in Wise’s work, and a starting point
for the proof of the virtual Haken conjecture. We
replace Wise’s cubical small-cancellation theory
with hyperbolic Dehn filling techniques, and a
quasiconvex combination theorem. This is joint work
with Daniel Groves and Jason Manning.

Mladen Bestvina, University of Utah

The geometry of Out(Fn), from Thurston to today and beyond

The study of the geometry of free group automorphisms began
when Thurston proved the Bounded Cancellation Lemma. Today the study
focuses on the large scale geometry of Out(Fn) and of
Culler-Vogtmann’s Outer space, modeled on the success of the
Masur-Minsky theory that explains the large scale geometry of mapping
class groups and Teichmüller space. The last several years have seen
a substantial progress, due to several people but in particular to
Algom-Kfir, Bestvina, Feighn, Handel and Mosher, but the Out(Fn)
theory is still lagging behind. I will review the main results and
indicate some of the challenges for the future.

Michel Boileau, Aix-Marseille Université

Cyclic branched coverings of knots and characterization of S3

A classical way to construct some closed orientable 3-manifolds is by
taking finite coverings of the 3-sphere S3 branched along knots.

Let call hyperelliptic rotation a periodic diffeomorphism of a closed
orientable 3-manifold corresponding to the covering translation
of a cyclic covering of S3 branched along a knot. A well-known
property of the standard sphere S3 is to admit hyperelliptic
rotations
of any order. In the other hand, due to Thurston’s orbifold theorem,
it is known that on a closed 3-manifold which is not homeomorphic to
S3,
the order of a hyperelliptic rotation, if it exists, is bounded by a
constant depending on the manifold.
In this talk we combine Thurston’s orbifold theorem and results from
the classification of finite simple groups to show that the
orientation preserving
diffeomorphism group of a closed irreducible 3-manifold which is not
homeomorphic to S3 contains at most 6 conjugacy classes of cyclic
subgroups
generated by a hyperelliptic rotation of prime odd order. In
particular a closed 3-manifold is homeomorphic to S3 iff it is a p-fold cyclic covering
of S3 branched along a knot for 7 distinct prime odd numbers p.
At the moment we do not know examples of a closed 3-manifold, other
than S3, which
is a p-fold cyclic covering of S3 branched along a knot for more
than 3 distinct odd prime numbers; for integral homology 3-spheres
3 is known to be
the maximal number.

This is a joint work with Clara Franchi, Mattia Mecchia, Luisa
Paoluzzi and Bruno Zimmermann.

Danny Calegari, University of Chicago

Random groups, diamonds and glass

In every dimension d, there are several infinite families of
convex cocompact Coxeter groups of isometries of hyperbolic d-space:
the superideal reflection groups. A random group at any density less
than a half (or in the few relators model) contains quasiconvex
subgroups commensurable with some member of any such family, with
overwhelming probability. This builds on and generalizes previous work
of Calegari-Walker and Calegari-Wilton.

Kelly Delp, Ithaca College

Playing with surfaces: spheres, monkey pants, and zippergons

I will describe a process, inspired by clothing design, of
smoothing an octahedron into a round sphere. This process was adapted
to build many surfaces out of paper and craft foam. The pattern pieces
for the surfaces were designed using a dynamic Mathematica notebook,
and cut using a digital cutter. This project was joint with Bill
Thurston.

Nathan Dunfield, University of Illinois at Urbana-Champaign

Practical computation with hyperbolic 3-manifolds

Computer experiment played an important motivational role in
Thurston’s work, including the formulation of his (now proved)
Geometrization Conjecture. In this talk, I will demonstrate the
computer program SnapPy for studying hyperbolic structures on
3-manifolds, which is based on ideas introduced by Thurston in his
revolutionary lecture notes. I will start with a brief tutorial on
the mechanics of using SnapPy and then devote most of the talk to
demonstrating what it can compute about hyperbolic 3-manifolds. I
will also give some interesting conjectures that can be studied with
SnapPy, and end by discussing
the future role computer experiment might play in 3-dimensional topology.

SnapPy is developed jointly with Marc Culler, Jeff Weeks, and many
others, and is freely available for all types of computers at snappy.computop.org

Benson Farb, University of Chicago

Homology, representation theory, and Bill

In the past few years I’ve been working with Thomas Church and Jordan Ellenberg (with Rohit Nagpal joining in as well) on building a theory we call “Representation stability/FI-modules,” whereby various sequences of spaces stabilize when viewed through the lens of representation theory. We’ve applied and connected this to topology (e.g., the cohomology of configuration spaces), arithmetic groups (e.g., cohomology of congruence subgroups), number theory (counting points on varieties, in particular polynomials over Fq), algebraic combinatorics (diagonal co-invariant algebras), and more.

In thinking about what to talk about at this conference, I wondered how to connect this stuff to Bill. I looked back over the past few years and saw: braids, polynomials, homology, representations, computer experiments — this had Bill written all over it! But more importantly, I realized that our entire project was infused with the Thurstonian viewpoint. In this talk I will describe some of this work.

Étienne Ghys, École Normale Supérieure de Lyon

Foliations: What's next after Thurston?

William Thurston had a fundamental impact on foliation theory in the 1970’s.
In this survey talk, I would like to give a rough outline of his contribution, in particular on the Godbillon-Vey invariant.
I’ll explain some recent results and focus on a few open problems.

Rick Kenyon, Brown University

Discrete analytic functions and integrability

We discuss tilings with squares, tilings with dominos,
discrete analytic functions, integrability, circle patterns
and the connections between these subjects.

François Labourie, Université Paris-Sud Orsay

Minimal surfaces and the complex geometry of Hitchin components

Teichmüller theory is a fascinating interplay between
complex geometry and topology: Teichmüller space is both a space of
complex structures on a surface S and a connected component
χ (PSL(2,R)) of representations of π1(S) in
PSL(2,R). For a general (simple real split) group
G, a similar connected component χ (G), called
the Hitchin component and its Thurstonian dynamical geometry has
attracted a lot of attention recently. Yet, the complex interpretation
of the Hitchin component is largely unknown (although conjectured). In
this talk, I will explain the complex interpretation of Hitchin
components for rank 2 groups as a space of pairs (J,q) where J is
a complex structure on S and q a holomorphic differential of an
order depending on the group G. I will use and explain
Hitchin-Corlette major results on non abelian Hodge theory.

Vlad Markovic, University of Cambridge

Homology of curves and surfaces in hyperbolic 3-manifolds

I will talk about my recent work with Yi Liu (building on the
previous work with Jeremy Kahn). We show that every second homology class
in a closed hyperbolic 3-manifold can be realised by a nearly geodesic
immersed surface, answering a question that Thurston asked two years ago.
Also, I will discuss our result that every homologically trivial curve
bounds a nearly geodesic surface and state some of its corollaries.

Dusa McDuff, Barnard College

Thurston's contributions to contact and symplectic geometry

This talk aims to describe some of Thurston’s contributions to contact
and symplectic geometry, and then to show how his ideas have been
developed.
His contributions include
some geometric constructions, as well as the definition of the
Thurston-Bennequin invariant. These mostly date from the 70s,
though the Eliashberg-Thurston theory of confoliations was developed
in the 90s.

Curtis McMullen, Harvard University

Dynamics and algebraic integers: Perspectives on Thurston's last theorem

No abstract available.

John Milnor, Stony Brook University

Hyperbolic component boundaries

When studying polynomial or rational maps in one complex variable, the easiest to understand are those with connected Julia set, and with the property that the orbit of every critical point converges towards an attracting periodic orbit. Any connected component of such maps, within the space of suitably normalized maps of fixed degree n ≥ 2, will be called a hyperbolic component. The talk will try to justify my belief that there is a basic dichotomy. The topological boundary of such a hyperbolic component must be either:

semi-algebraic, defined by polynomial equalities and inequalities,

or else not locally connected.

Yair Minsky, Yale University

Relative and absolute bounds on skinning maps

Thurston’s skinning map, together with his Bounded Image Theorem, are
a crucial ingredient in his proof of hyperbolization for Haken
3-manifolds. A better quantitative understanding of this map can help
us connect topological descriptions of 3-manifolds to their geometric
features. I’ll discuss some partial results in this direction,
including joint work with Kent and with Brock, Bromberg and Canary.

Yi Ni, California Institute of Technology

Genus minimizing knots in rational homology spheres

Before his work on geometrization, Thurston introduced the Thurston norm
of 3-manifolds and related it to taut foliations. Based on his work,
it was proved that Heegaard Floer homology determines the Thurston norm.
In this talk, we will discuss an analogue of Thurston norm for rational
homology spheres, and its lower bound from Heegaard Floer homology. Such a
bound is related to problems like lens space surgery, one-sided Heegaard splitting, and complexity of 3-manifolds. This talk is based on joint work
with Zhongtao Wu and with Josh Greene.

Alan Reid, University of Texas at Austin

Arithmetic hyperbolic manifolds

Thurston had a longstanding interest in understanding
geometric and topological properties of arithmetic hyperbolic
manifolds. For example in his 1982 Bulletin Article, as Qn 19
of the problem list, Thurston posed: “Find topological and geometric
properties of quotient spaces of arithmetic subgroups of PSL(2,C). These
manifolds often seem to have special beauty.”

This talk will take up this theme, report on some recent work, as well
as describe future directions.

Richard Schwartz, Brown University

A smorgasbord of computer demos

I’ll demonstrate some of the graphical user interfaces
I’ve written over the years to help understand some geometrically
defined dynamical systems — billiards, polygon exchange transformations,
and polygon iterations.

Mitsuhiro Shishikura, Kyoto University

Thurston's theorems in complex dynamics

Thurston made several important contributions to
complex dynamics, such as the topological characterization of
rational maps among self branched coverings of 2-spheres and
the theory of laminations for polynomials. The former
was especially important in giving a systematic way to construct rational maps
from given topological/combinatorial data. Also its essential use of
Teichmüller theory changed the way we think about complex dynamics.
I will try to explain the role of this result in complex dynamics and
related open questions.

Tan Lei, Université d'Angers

Core entropy of polynomials

I will present Bill Thurston’s work on core entropy of polynomials.
The entropy for a polynomial map of degree d acting on its entire
Julia set is always log d.
What’s tricky is to figure out the best definition that filters away
the action on the tips of the Julia set, and consider the action on
the “core part” only.

For a real polynomial the “core part” is simply the real trace of the
Julia set, as studied by Milnor-Thurston
many years ago. For a complex polynomial a natural candidate is the
Hubbard tree (the convex hull of the postcritical set). But this tree
does not always exist, and even when it exists, it’s not always easy to
compute the tree and the
induced action.

Bill found a way to define in a combinatorial way the “core” directly
from a combinatorial description (what he called primitive major) of
the polynomial. He then figured out an effective algorithm that
computes the core entropy. He also gave a beautiful characterization
of the parameter space of these primitive majors.
Many questions remain open in this new and fascinating field. I will
present some of them.

Jeff Weeks

The shape of space

When we look out on a clear night, the universe seems infinite. Yet this
infinity might be an illusion. During the first half of the presentation, computer games
will introduce the concept of a multiconnected universe. Interactive 3D graphics will then
take the viewer on a tour of several possible shapes for space, and we’ll see how recent
satellite data provide tantalizing clues to the true shape of our universe. Finally, we'll see
what Bill Thurston’s pioneering discoveries say about the curvature of space, using
fleece
surfaces to illustrate his main idea. The only prerequisites for this talk are curiosity and
imagination. For middle school and high school students, people interested in astronomy,
and all members of the Ithaca community.

Anna Wienhard, University of Heidelberg and Princeton University

Geometric structures and representation varieties

I will describe some developments in the study of locally homogeneous geometric structures and related subsets of representation varieties of hyperbolic groups into Lie groups of higher rank. I will also discuss several open questions.

Dani Wise, McGill University

Mixed 3-manifolds are virtually special

Let M be a compact irreducible 3-manifold that is neither a
graph manifold nor a hyperbolic
manifold. We prove that the fundamental group of M is virtually special.
This is joint work with Piotr Przytycki.

Anton Zorich, Université Paris 7 Jussieu

Lyapunov exponents of the Hodge bundle and diffusion in periodic billiards

Asymptotic behavior of leaves of a measured foliation on a Riemann
surface is governed by the mean monodromy of the Hodge bundle along
the associated trajectory of the Teichmüller geodesic flow in the
moduli space. As a consequence, recent progress in the study of the
Teichmüller flow (inspired by the fundamental work of A. Eskin and
M. Mirzakhani) and in the study of the Lyapunov exponents of the
Hodge bundle along this flow leads to new results on measured
foliations on surfaces.

Following ideas of V. Delecroix, P. Hubert, and S. Lelièvre I will
show how to apply this technique to description of the diffusion of
billiard trajectories in the plane with periodic polygonal obstacles.
The results presented in the talk are obtained in collaboration with
J. Athreya, V. Delecroix, A. Eskin, and M. Kontsevich.